Type about:config in the address bar, search "update", set "app.update.auto" to false. Older versions used to have a menu checkbox for that... no more.

Firefox seems like it doesn't always want to translate and retain tweaks from the GUI menus, so while in about:config I like to carefully peruse and disable (with Google's help) numerous other auto update stuff with search terms like "update", "health", "crash", etc.

Try the above, and see if that helps calm down the background chatter...